1. Field of the Invention
The present invention relates to gravity gradiometry and, more particularly, to quadrupole gravity gradiometers.
2. Description of the Related Art
Measuring the gravitational field involves quantifying the gravitational forces acting on test masses. Gravimeter instruments measure the acceleration of gravity and are sensitive to linear accelerations. Doing measurements with an instrument carried by a non-inertial moving platform requires evaluating the inertial forces acting on test masses and subtracting their effect from the measured signal. Also, platform induced forces are many orders of magnitude higher than the variations of acceleration of gravity measured and an appropriate suspension is required on order to reduce such forces to a level that can be handled by the dynamic range of the instrument. For that reason gravity gradiometer instruments are more useful, since they have low sensitivity to linear acceleration by their operating principle.
The gravity potential generated by mass m at distance r is
  Φ  =      G    ⁢          m      r      
where G=6.674×10−11 m3 kg−1 s−2 is the Newtonian constant of gravitation. The acceleration of gravity in that potential field is
  g  =                    (                              g            x                    ,                      g            y                    ,                      g            z                          )            T        =                            (                                                    ⅆ                Φ                                            ⅆ                x                                      ,                                          ⅆ                Φ                                            ⅆ                y                                      ,                                          ⅆ                Φ                                            ⅆ                z                                              )                T            .      
The tensor of the gravity acceleration gradient is
  Γ  =            (                                                  Γ              xx                                                          Γ              xy                                                          Γ              xz                                                                          Γ              yx                                                          Γ              yy                                                          Γ              yz                                                                          Γ              zx                                                          Γ              zy                                                          Γ              zz                                          )        =                  (                                                                              ⅆ                                      g                    x                                                                    ⅆ                  x                                                                                                      ⅆ                                      g                    x                                                                    ⅆ                  y                                                                                                      ⅆ                                      g                    x                                                                    ⅆ                  z                                                                                                                          ⅆ                                      g                    y                                                                    ⅆ                  x                                                                                                      ⅆ                                      g                    y                                                                    ⅆ                  y                                                                                                      ⅆ                                      g                    y                                                                    ⅆ                  z                                                                                                                          ⅆ                                      g                    z                                                                    ⅆ                  x                                                                                                      ⅆ                                      g                    z                                                                    ⅆ                  y                                                                                                      ⅆ                                      g                    z                                                                    ⅆ                  z                                                                    )            =                        (                                                                                                                ⅆ                      2                                        ⁢                    Φ                                                        ⅆ                                          x                      2                                                                                                                                                              ⅆ                      2                                        ⁢                    Φ                                                                              ⅆ                      x                                        ⁢                                          ⅆ                      y                                                                                                                                                              ⅆ                      2                                        ⁢                    Φ                                                                              ⅆ                      x                                        ⁢                                          ⅆ                      z                                                                                                                                                                                      ⅆ                      2                                        ⁢                    Φ                                                                              ⅆ                      y                                        ⁢                                          ⅆ                      x                                                                                                                                                              ⅆ                      2                                        ⁢                    Φ                                                        ⅆ                                          y                      2                                                                                                                                                              ⅆ                      2                                        ⁢                    Φ                                                                              ⅆ                      y                                        ⁢                                          ⅆ                      z                                                                                                                                                                                      ⅆ                      2                                        ⁢                    Φ                                                                              ⅆ                      z                                        ⁢                                          ⅆ                      x                                                                                                                                                              ⅆ                      2                                        ⁢                    Φ                                                                              ⅆ                      z                                        ⁢                                          ⅆ                      y                                                                                                                                                              ⅆ                      2                                        ⁢                    Φ                                                        ⅆ                                          z                      2                                                                                                    )                .            
Obviously, the order of differentiation is irrelevant, therefore the tensor is symmetric. It also has zero trace (the sum of diagonal elements is zero) since gravity obeys Laplace's equation in free space. Therefore only five components of the tensor are independent.
The gravity acceleration gradient is measured in Eötvös units (E) defined as 1 E=10−9 s−1. Current commercial gradiometers have a sensitivity of a few Eötvös units.